Date: Mar 13, 2013 3:09 PM
Author: Zaljohar@gmail.com
Subject: Re: Reducing Incomparability in Cardinal comparisons

On Mar 12, 10:46 pm, Zuhair <zaljo...@gmail.com> wrote:
> On Mar 12, 2:49 pm, Zuhair <zaljo...@gmail.com> wrote:
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> > On Mar 11, 11:17 pm, Zuhair <zaljo...@gmail.com> wrote:
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> > > Let x-inj->y stands for there exist an injection from x to y and there
> > > do not exist a bijection between them; while x<-bij-> means there
> > > exist a bijection between x and y.

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> > > Define: |x|=|y| iff  x<-bij->y
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> > > Define: |x| < |y| iff  x-inj->y Or Rank(|x|) -inj-> Rank(|y|)
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> > > Define: |x| > |y| iff |y| < |x|
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> > > Define: |x| incomparable to |y| iff ~|x|=|y| & ~|x|<|y| & ~|x|>|y|
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> > > where |x| is defined after Scott's.
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> > > Now those are definitions of what I call "complex size comparisons",
> > > they are MORE discriminatory than the ordinary notions of cardinal
> > > comparisons. Actually it is provable in ZF that for each set x there
> > > exist a *set* of all cardinals that are INCOMPARABLE to |x|. This of
> > > course reduces incomparability between cardinals from being of a
> > > proper class size in some models of ZF to only set sized classes in
> > > ALL models of ZF.

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> > > However the relation is not that natural at all.
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> > > Zuhair
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> > One can also use this relation to define cardinals in ZF.
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> > |x|={y| for all z in TC({y}). z <* x}
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> > Of course <* can be defined as:
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> > x <* y iff [x -inj->y Or
> > Exist x*. x*<-bij->x & for all y*. y*<-bij->y -> rank(x*) in
> > rank(y*)].

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> > Zuhair
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> All the above I'm sure of, but the following I'm not really sure of:
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> Perhaps we can vanquish incomparability altogether
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> If we prove that for all x there exist H(x) defined as the set of all
> sets hereditarily not strictly supernumerous to x. Where strict
> subnumerousity is the converse of relation <* defined above.
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> Then perhpas we can define a new Equinumerousity relation as:
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> x Equinumerous to y iff H(x) bijective to H(y)
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> Also a new subnumerousity relation may be defined as:
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> x Subnumerous* to y iff H(x) injective to H(y)


Better would be

x Subnumerous* to y iff H(x) <* H(y)

however still this won't concur incomparability completely

However if we define recursively H_n(x) then we can define
the above relations after those. However still incomparability
would persist, although the above is still a strong approach
against incomparability.
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> This might resolve all incomparability issues (I very highly doubt
> it).
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> Then the Cardinality of a set would be defined as the set of all sets
> Equinumerous to it of the least possible rank.
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> A Scott like definition, yet not Scott's.
>
> Zuhair